One mole of an ideal gas initially kept in a cylinder at pressure 1 MPa and temperature `27^(@)C` is made to expand until its volume is doubled.
(a) How much work is done if the expansion is (i) adiabatic (ii) isobaric (iii) isothermal?
(b) Identify the processes in which workdone is least and is maximum.
(c) Show each process on a PV diagram.
(d) Name the processes in which the heat transfer is maximum and minimum.
`("Take "gamma = 5//3` and `R = 8.3 J mol^(-1) K^(-1))`

Explanation

(a) (i) In an adiabatic process the work done by the system is
`W_("adia")=(muR)/(gamma-1)[T_(i)-T_(f)]`
To find the temperature `T_(f)`, we can use adiabatic equation of state.
`T_(f)V_(f)^(gamma-1)=T_(i)V_(i)^(gamma-1)`
`T_(f)=T_(i)((V_(i))/(V_(f)))^(gamma-1)=300xx((1)/(2))^((2)/(3))=0.63xx300K=189.8K`
`W=1xx8..3xx(3)/(2)(300-189.8)=1.37kJ`
(ii) In an isobaric process the work done by the system
`W=PDeltaV=P(V_(f)-V_(i)) and V_(f) =2V_(i)` so `W=2PV_(i)`
To find `V_(i)`, we can use the ideal gas for initial state, `P_(i)V_(i)=RT_(i)`
`V_(i)=(RT_(i))/(P_(i))=8.3xx(300)/(1)xx10^(-6)=24.9xx10^(-4)m^(3)`
The work doen during isobaric process,
`W=2xx10^(6)xx24.9xx10^(-4)=4.9kJ`
(iii) In an isothermal process the work doen by dthe system,
`W=muRT ln ((V_(f))/(V_(i)))`
In an isothermal process the initial room temperature is constant.
`W=1xx8.3xx300xxln (2)=1.7 kJ`
(b) Comparing all three processes, we see that the work done in the isobaric process is the greatest, and work done in the adiabatic process is the least least
(c) The PV diagram is shown in the Figure.
The area under the curve AB=Work done during the isobaric process
The area under the curve AC=Work done during the isothermal process
The area under the curve AD= Work done during the adiabatic process
From the PV diagram the area under the curve AB is more, implying that the work done in isobaric process is highest and work done in adiabatic process is least.
(d) In an adiabatic process no heat enters into the system or leaves from the system. In an isobaric process the work done is more so heat supplied should be more compared to an isothermal process.